§ 9.5. Integral Representations

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§9.5(i)Real Variable
§9.5(ii)Complex Variable

§ 9.5(i). Real Variable

Notes:: See Miller (1946, p. B17) and Olver (1997b, p. 103).
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9.5.1 $\mathrm{Ai}\!\left(x\right)=\frac{1}{\pi}\int _{0}^{\infty}\cos\!\left(\tfrac{1}{3}t^{3}+xt\right)dt.$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $x$ : real variable
A&S Ref:: 10.4.32 (in different form)
Permalink:: http://dlmf.nist.gov/9.5.E1
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9.5.2 $\mathrm{Ai}\!\left(-x\right)=\frac{x^{{\ifrac{1}{2}}}}{\pi}\int _{{-1}}^{\infty}\cos\!\left(x^{{\ifrac{3}{2}}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)dt,$ $x>0$ .

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $x$ : real variable
A&S Ref:: 10.4.32 (in different form)
Permalink:: http://dlmf.nist.gov/9.5.E2
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9.5.3 $\mathrm{Bi}\!\left(x\right)=\frac{1}{\pi}\int _{0}^{\infty}\exp\!\left(-{\tfrac{1}{3}}t^{3}+xt\right)dt+\frac{1}{\pi}\int _{0}^{\infty}\sin\!\left(\tfrac{1}{3}t^{3}+xt\right)dt.$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function and $x$ : real variable
A&S Ref:: 10.4.33
Referenced by:: §9.12(vii)
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See also (9.10.19), (9.11.3), (Ch.36), and Vallée and Soares (2004, §2.1.3).

§ 9.5(ii). Complex Variable

Notes:: For (9.5.4) see Olver (1997b, p. 53). For (9.5.5) combine (9.2.10) and (9.5.4). For (9.5.6) see Reid (1995). For (9.5.7) see Copson (1963). (9.5.8) follows from the first of (9.6.2) and (Ch.10).
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9.5.4 $\mathrm{Ai}\!\left(z\right)=\frac{1}{2\pi i}\int _{{\infty e^{{-\pi i/3}}}}^{{\infty e^{{\pi i/3}}}}\exp\!\left(\tfrac{1}{3}t^{3}-zt\right)dt,$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $z$ : complex variable
Referenced by:: §9.14, §9.17(iii), §9.5(ii)
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9.5.5 $\mathrm{Bi}\!\left(z\right)=\frac{1}{2\pi}\int _{{-\infty}}^{{\infty e^{{\pi i/3}}}}\exp\!\left(\tfrac{1}{3}t^{3}-zt\right)dt+\dfrac{1}{2\pi}\int _{{-\infty}}^{{\infty e^{{-\pi i/3}}}}\exp\!\left(\tfrac{1}{3}t^{3}-zt\right)dt.$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function and $z$ : complex variable
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E5
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9.5.6 $\mathrm{Ai}\!\left(z\right)=\frac{\sqrt{3}}{2\pi}\int _{0}^{\infty}\exp\!\left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)dt.$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $z$ : complex variable
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E6
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9.5.7 $\mathrm{Ai}\!\left(z\right)=\frac{e^{{-\zeta}}}{\pi}\int _{0}^{\infty}\exp\!\left(-z^{{\ifrac{1}{2}}}t^{2}\right)\cos\!\left(\tfrac{1}{3}t^{3}\right)dt,$ $|\mathrm{ph}z|<\pi$ .

Defines:: $\zeta$ : change of variable
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $z$ : complex variable
Referenced by:: §9.5(ii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E7
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9.5.8 $\mathrm{Ai}\!\left(z\right)=\frac{e^{{-\zeta}}\zeta^{{\ifrac{-1}{6}}}}{\sqrt{\pi}(48)^{{\ifrac{1}{6}}}\Gamma\!\left(\frac{5}{6}\right)}\int _{0}^{\infty}e^{{-t}}t^{{-\ifrac{1}{6}}}\left(2+\frac{t}{\zeta}\right)^{{-\ifrac{1}{6}}}dt,$ $|\mathrm{ph}z|<\frac{2}{3}\pi$ .

Defines:: $\zeta$ : change of variable
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Referenced by:: §9.17(iii), §9.5(ii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E8
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In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{{\ifrac{3}{2}}}$ .

§ 9.5. Integral Representations

Contents

§ 9.5(i). Real Variable

§ 9.5(ii). Complex Variable